Walter Mazzarri's men came into the game in fine form, unbeaten in their first five competitive fixtures without conceding a goal and having thrashed Sassuolo 7-0 last weekend.
But they fell behind to Franco Vazquez's early strike and were second best in the first half before Mateo Kovacic came up with something from nothing to get them back into it before half-time.
The Milanese outfit twice thought they had taken the lead, either side of the break, but were denied on both occasions by an offside flag, while the hosts were also frustrated by a disallowed goal in the dying embers of the match - a decision that coach Giuseppe Iachini protested so fiercely that he was banished from the touchline.
The result leaves Palermo looking for their first league victory since their return to the top flight and Inter with five points from three games.
Inter's confidence was clear from the first whistle as Danilo D'Ambrosio, who scored the winner against Dnipro in the UEFA Europa League on Thursday, drove down the right from the kick-off and won a dangerous free-kick.
That setback sapped the self-assurance from Inter's play and they struggled to find their rhythm, uncomfortable in possession all over the pitch.
Mazzarri's charges played themselves into trouble time and again, and would have been two down had Michel Morganella made any contact with Achraf Lazaar's teasing left-wing cross in the 19th minute.
Vidic then thought he had atoned for his error and completed an unlikely turnaround two minutes later but his header from a Fredy Guarin free-kick was ruled out, with Pablo Daniel Osvaldo in an offside position.
It was Palermo's turn to feel aggrieved in the 88th minute when substitute Andrea Belotti beat Handanovic on the break, only to have been judged to have fouled Yuto Nagatomo in the build-up.
